Tracing the history of medieval astronomy
Now that we’ve worked out a distance to the sun and moon, as well as the angular diameter, we can put these together to determine the sizes which is what Ptolemy seeks to do in this chapter. Fortunately, we don’t need a new diagram and can simply make reference to the one we used in the last post:
Continue reading “Almagest Book V: Size of the Sun, Moon, and Earth”
Ptolemy starts the chapter reminding us that we found distance to the moon at its maximum distance was $64;10^p$. This was comprised of the distance from earth the mean distance at syzygy being $59^p$ and the radius of the epicycle being an addition $5;10$. Using that,
let us see the size of the sun’s distance which results. Continue reading “Almagest Book V: Distance to the Sun”
In order to determine the relationship between the true distance to the sun and moon, one of the key pieces of information we’ll need is their apparent angular diameters at syzygy. I’ve mentioned several time that both are around half a degree. For this to work, we’ll need to be more accurate than that, but measuring small angles like this is especially tricky. Ptolemy mentions a few ways astronomers prior to him tried to tackle the issue which include
measuring [the flow of] water1 or by the time [the sun and moon] take to rise at the equinox.
However, he rejects these stating that they are not sufficiently accurate. Instead, he states he used a dioptra which is a surveying instrument. While he doesn’t get into the detail of its construction or use, he does give a summary of some of his key findings:
First, he states that the sun’s apparent diameter does not appear to change, but the moon’s apparent diameter does. According to Ptolemy, it has the same angular distance as the sun only when it is at its maximum distance2 which he states disagrees with his predecessors who claimed that the moon’s diameter matched that of the sun only at mean distance. Ptolemy’s position is easily refuted as the existence of annular eclipses by necessitates a smaller angular diameter of the moon than the sun.
The second conclusion Ptolemy gives is that the angular diameters he has determined are “considerably smaller than those traditionally accepted.” Although he doesn’t give any of those previous values, Toomer notes that Hipparchus had a value of “a six hundred and fiftieth of its circle” which is about $0;33,14º$. Ptolemy also doesn’t give away his figure just yet either, but does indicate that it didn’t actually come from use of the dioptra he just mentioned, but “on certain lunar eclipses.”
So in this post, we’ll explore that method, but as a warning, while Toomer praises this section as “elegant and theoretically correct” this section has many problems. Continue reading “Almagest Book V: Angular Diameter of the Moon and Earth’s Shadow at Apogee During Syzygy”
Now that we’ve worked out the distance to the moon at the time of the observation, we can put this information back into our lunar model diagram to work out the true scale. We’ll begin with a drawing of our lunar model at the time depicted:
Continue reading “Almagest Book V: Scale of the Lunar Model”
Now that we’ve demonstrated a parallax of $1;07º$ for the moon, we can use that do determine a distance to the moon. As a forewarning, some of the math may seem suspect here, so I’ll do my best to explain it.
To being, let’s start off with a new drawing:
Continue reading “Almagest Book V: Calculation of Lunar Distance”
So far in this book, we’ve refined our lunar model, shown how to use it to calculate the lunar position3, discussed a new instrument suitable for determining lunar parallax, as well as an example of the sort of observation necessary to make the calculation.
Now, Ptolemy walks us through an example of how to calculate the lunar distance using an example entirely unrelated to the one we saw in the last post.
In the twentieth year of Hadrian, Athyr [III] $13$ in the Egyptian calendar [135 CE, Oct. $1$4], $5 \frac{5}{6}$ equinoctial hours after noon, just before sunset, we observed the moon when it was on the meridian. The apparent distance of its center from the zenith, according to the instrument, was $50 \frac{11}{12}º$. For the distance [measured] on the thin rod was $51 \frac{7}{12}$ of the $60$ subdivisions into which the radius of revolution had been divided, and a chord of that size subtends an arc of $50 \frac{11}{12}º$.
Continue reading “Almagest Book V: Calculation of Lunar Parallax”
In the last post we followed along as Ptolemy discussed the construction and use of his parallactic instrument, which he would use to measure the lunar parallax. To do so, Ptolemy waited for the moon to
be located on the meridian, and near the solstices on the ecliptic, since at such situations, the great circle through the poles of the horizon and the center of the moon very nearly coincides with the great circle through the poles of the ecliptic, along which the moon’s latitude is taken.
That’s pretty dense, so let’s break it down with some pictures, First, let’s draw exactly what Ptolemy has described above:
Continue reading “Almagest Book V: Lunar Parallactic Observations”
The primary instrument I’ve used for my observing is an astronomical quadrant. That instrument is designed primarily to measure the angular distance of an object above the horizon5, otherwise known as its altitude. However, this isn’t the only instrument good for this sort of thing. Brahe’s Astronomiae Instauratae Mechanica is filled with instruments that essentially fill this same purpose, but in different ways.
One design, he describes as a “parallactic instrument” but it was also known as a triquetrum in period. This design dates back to Ptolemy and is described in Chapter $12$. Here’s a drawing of it from Toomer:
Continue reading “Almagest Book V: The Parallactic Instrument”
Chapter 11 of Book 5 is one of those rare chapters that’s blessedly free of any actual math. Instead, Ptolemy gives an overview of the problem of lunar parallax, stating that it will need to be considered because “the earth does not bear the ratio of a point to the distance of the moon’s sphere.” In other words, the ratio of the diameter of the earth to the distance of the moon isn’t zero.
However, this does pose an interesting question. We’ve previously given the radius of the eccentre as $49;41^p$, but we haven’t given the radius of the Earth in the same units. Thus, how can this ratio even be taken to know this? Continue reading “Almagest Book V: Lunar Parallax”
In the last post, we looked at how much the total equation of anomaly would change during syzygy due to the eccentre we added to the lunar model in this book, when the moon was at its greatest base equation of anomaly. As Ptolemy told us, it wasn’t much. However, there was a second effect that can also change the equation of anomaly, which was based on where we measure the movement around the epicycle from. Namely, the mean apogee instead of the true apogee. This has its maximum effect when the moon is near apogee or perigee so in this post, we’ll again quantify how much.
Let’s start off by building our diagram:
Continue reading “Almagest Book V: The Difference at Syzygies – Lunar Apogee and Perigee”